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B Why differential of e^x is special?

  1. Nov 5, 2016 #1
    I learnt that differential of e^x is same but whats so special about it? What makes is so special as it seems like a normal function to me other than the fact that e= sum of series of reciprocal of factorial numbers. What i want to ask is if e^x differential is e^x then do this rule apply to b^x too? Where b is a positive base. Why or why not?
     
  2. jcsd
  3. Nov 5, 2016 #2

    Math_QED

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    Homework Helper

    No ##\frac{db^x}{dx} = \log_e (b) b^x##.
    So we get the desired formula when ##b = e##

    When you would look at differentiation as an operation in a function space , ##e^x## would be a neutral element for this operation.
     
    Last edited: Nov 5, 2016
  4. Nov 5, 2016 #3

    Mark44

    Staff: Mentor

    Actually, what you're asking about is the derivative of ex, not the differential of ex.
     
  5. Nov 5, 2016 #4

    fresh_42

    Staff: Mentor

    One can ask which functions ##f : \mathbb{R} \longrightarrow \mathbb{R}## satisfy ##\frac{d}{dx}f(x) = f(x)##. The solution is ##x \mapsto c \cdot e^x##.
    So ##x \mapsto e^x## is the solution that additionally satisfies the condition ##f(0) = 1##.
    This is one way to define the exponential function.

    Others are a limit or a version of the series you mentioned.

    As you also mentioned ##g(x) = b^x##, there is one thing they have in common: both satisfy ##g(x+y) = g(x) + g(y)##.
    But as @Math_QED has said, ##\frac{d}{dx} g(x) = \log(b) \cdot g(x) \neq g(x)## if the basis isn't ##e##.
     
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